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Theorem nexdv 1866
Description: Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 13-Jul-2020.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
Hypothesis
Ref Expression
nexdv.1 (𝜑 → ¬ 𝜓)
Assertion
Ref Expression
nexdv (𝜑 → ¬ ∃𝑥𝜓)
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem nexdv
StepHypRef Expression
1 ax-5 1841 . 2 (𝜑 → ∀𝑥𝜑)
2 nexdv.1 . 2 (𝜑 → ¬ 𝜓)
31, 2nexdh 1790 1 (𝜑 → ¬ ∃𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841
This theorem depends on definitions:  df-bi 197  df-ex 1702
This theorem is referenced by:  sbc2or  3431  csbopab  4973  relimasn  5451  csbiota  5843  canthwdom  8429  cfsuc  9024  ssfin4  9077  konigthlem  9335  axunndlem1  9362  canthnum  9416  canthwe  9418  pwfseq  9431  tskuni  9550  ptcmplem4  21764  lgsquadlem3  25002  umgredgnlp  25930  dfrdg4  31692
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